Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves

TL;DR

This paper introduces explicit isogenies that transfer the discrete logarithm problem from hyperelliptic genus 3 Jacobians to non-hyperelliptic ones, where DLP is easier to solve, revealing vulnerabilities in certain cryptographic systems.

Contribution

It provides explicit formulae for isogenies with kernel $(\ZZ/2\ZZ)^3$ for hyperelliptic genus 3 curves and demonstrates their practical application in reducing DLP instances.

Findings

Explicit isogenies exist for a significant fraction of hyperelliptic genus 3 curves.

The reduction from hyperelliptic to non-hyperelliptic Jacobians is efficient and explicit.

Approximately 18.57% of hyperelliptic genus 3 curves over finite fields can be reduced using these isogenies.

Abstract

We describe the use of explicit isogenies to translate instances of the Discrete Logarithm Problem (DLP) from Jacobians of hyperelliptic genus 3 curves to Jacobians of non-hyperelliptic genus 3 curves, where they are vulnerable to faster index calculus attacks. We provide explicit formulae for isogenies with kernel isomorphic to $(\ZZ/2\ZZ)^3$ (over an algebraic closure of the base field) for any hyperelliptic genus 3 curve over a field of characteristic not 2 or 3. These isogenies are rational for a positive fraction of all hyperelliptic genus 3 curves defined over a finite field of characteristic $p > 3$. Subject to reasonable assumptions, our constructions give an explicit and efficient reduction of instances of the DLP from hyperelliptic to non-hyperelliptic Jacobians for around 18.57% of all hyperelliptic genus 3 curves over a given finite field. We conclude with a discussion on extending these ideas to isogenies with more general kernels. A condensed version of this work appeared in the proceedings of the EUROCRYPT 2008 conference.